🔲 Algebra Solver

Completing the Square Calculator - Step-by-Step

Solve quadratic equations by completing the square with clear step-by-step calculations.

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How to Use & Educational Guide

In-depth guide for Completing the Square Calculator

Educational Overview

How to Complete the Square?

Completing the square is an algebraic method used to rewrite a quadratic expression in the form of a perfect square. It is commonly used to solve quadratic equations, find the zeros of a quadratic, and convert a parabola into vertex form. A quadratic such as $x^2 + bx + c$ can be transformed into a form containing $(x + p)^2$.

The Completing the Square Formula

For an expression where the coefficient of $x^2$ is $1$, take half of the coefficient of $x$, square it, and add the result:

$$x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2$$

When solving an equation, the same quantity must be added to both sides to preserve equality.

How to Solve a Quadratic Equation by Completing the Square?

To solve a quadratic equation by completing the square, transform the equation into a perfect square and then use square roots to find the solutions.

Step 1 (Write in Standard Form): Arrange the quadratic equation as $ax^2 + bx + c = 0$.
Step 2 (Make the $x^2$ Coefficient 1): If $a \ne 1$, divide the equation by $a$ so that the coefficient of $x^2$ becomes $1$.
Step 3 (Move the Constant): Move the constant term to the opposite side, leaving the $x^2$ and $x$ terms together.
Step 4 (Take Half and Square): Take half of the coefficient of $x$, square it, and add this value to both sides.
Step 5 (Form the Perfect Square): Rewrite the left side as a squared binomial such as $(x + p)^2$.
Step 6 (Take Square Roots): Apply the square-root property to both sides and include both the positive and negative possibilities.
Step 7 (Solve for $x$): Isolate $x$ to obtain the final roots or solutions.

Completing the Square Calculator Examples

Example 1: Solve by Completing the Square: $x^2 + 6x + 5 = 0$

• Step 1 (Move the constant): $x^2 + 6x = -5$.

• Step 2 (Take half of 6 and square it): $\left(\frac{6}{2}\right)^2 = 3^2 = \mathbf{9}$.

• Step 3 (Add 9 to both sides): $x^2 + 6x + 9 = -5 + 9$.

• Step 4 (Form the perfect square): $(x + 3)^2 = \mathbf{4}$.

• Step 5 (Take square roots): $x + 3 = \pm\sqrt{4} = \pm 2$.

• Step 6 (Solve): $x = -3 + 2 = \mathbf{-1}$ or $x = -3 - 2 = \mathbf{-5}$.

👉 Roots: $x = -1, \quad x = -5$

Example 2: Solve $2x^2 + 8x + 6 = 0$ by Completing the Square

• Step 1 (Divide by 2): $x^2 + 4x + 3 = 0$.

• Step 2 (Move the constant): $x^2 + 4x = -3$.

• Step 3 (Complete the square): $\left(\frac{4}{2}\right)^2 = 4$, so add 4 to both sides.

• Step 4: $x^2 + 4x + 4 = -3 + 4 \implies (x + 2)^2 = 1$.

• Step 5 (Take square roots): $x + 2 = \pm 1$.

• Step 6 (Solve): $x = -1$ or $x = -3$.

👉 Roots: $x = -1, \quad x = -3$

Example 3: Convert to Vertex Form: $y = x^2 - 8x + 12$

• Step 1 (Group the $x$ terms): $y = (x^2 - 8x) + 12$.

• Step 2 (Complete the square): Half of $-8$ is $-4$, and $(-4)^2 = 16$.

• Step 3 (Add and subtract 16): $y = (x^2 - 8x + 16) + 12 - 16$.

• Step 4 (Write vertex form): $y = (x - 4)^2 - 4$.

👉 Vertex: $(h, k) = (4, -4)$

Example 4: Solve $x^2 - 4x - 5 = 0$ by Completing the Square

• Step 1 (Move the constant): $x^2 - 4x = 5$.

• Step 2 (Take half of $-4$ and square it): $\left(\frac{-4}{2}\right)^2 = 4$.

• Step 3 (Add 4 to both sides): $x^2 - 4x + 4 = 5 + 4$.

• Step 4 (Form the perfect square): $(x - 2)^2 = 9$.

• Step 5 (Take square roots): $x - 2 = \pm 3$.

• Step 6 (Solve): $x = 5$ or $x = -1$.

👉 Roots: $x = 5, \quad x = -1$

Find Zeros by Completing the Square

The zeros of a quadratic are the values of $x$ that make the equation equal to zero. Completing the square can be used to find these zeros by converting the quadratic into a squared expression and then solving for $x$. For example, $x^2 - 4x - 5 = 0$ becomes $(x - 2)^2 = 9$, giving the zeros $x = 5$ and $x = -1$.

Why is Completing the Square Important?

📐 Parabolas & Vertex Form

Completing the square converts a quadratic into vertex form, making it easier to identify the vertex, axis of symmetry, and maximum or minimum value of a parabola.

⭕ Circle Equations

Completing the square for both $x$ and $y$ terms can transform a general circle equation into standard center-radius form: $(x-h)^2 + (y-k)^2 = r^2$.

Common Mistakes to Avoid

Forgetting to Add to Both Sides: When solving an equation, the value used to complete the square must be added to both sides to keep the equation balanced.
Using the Wrong Half: If the coefficient of $x$ is $6$, use $(6/2)^2 = 9$, not $6^2$.
Forgetting the ± Sign: From an equation such as $(x+3)^2 = 4$, the solutions are $x+3 = \pm2$, giving two possible roots.
Ignoring the Leading Coefficient: If the coefficient of $x^2$ is not $1$, first divide through by that coefficient when appropriate before completing the square.

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Frequently Asked Questions (FAQs)

Answers to common questions about this calculator

What is completing the square?

Completing the square is a method for rewriting a quadratic expression as a perfect square. It can be used to solve quadratic equations and find the vertex of a parabola.

How do you solve a quadratic equation by completing the square?

Write the equation in standard form, make the coefficient of x² equal to 1, move the constant, add (b/2)² to both sides, form a perfect square, take square roots, and solve for x.

What is the formula for completing the square?

For x² + bx, add (b/2)² to create the perfect square x² + bx + (b/2)² = (x + b/2)².

Can completing the square find the zeros of a quadratic?

Yes. Completing the square transforms a quadratic equation into a squared form that can be solved using square roots, giving the zeros or roots of the equation.